1 Introduction

When writing a paper using a method validated by this software, we recommend that you cite the software beyond and provide the raw data used as supplementary material for your paper.

To cite this software :

ValidR - Assay

2 Methods

2.1 Statistical analysis

All statistical were produced using the method described in Hubert, P. et al. 2007. Harmonization of strategies for the validation of quantitative analytical procedures. A SFSTP proposal - part III. J Pharm Biomed Anal 45: 82-96. And algorithm were checked using the data providied with the article.

The \(\beta\) value used for the calculation of the \(\beta\) expectation tolerance interval was set to 0.8 and the acceptance limits was fixed to 5 %.

The estimation of the parameters of the calibration curves was obtained by the ordinary least squares method (OLS) with R version 4.2.2 (2022-10-31) software [@R-base] and the r stats::lm() function was used. .

3 Analysis of response function (calibration curves)

3.1 Methods and Data

Response functions \(signal=f(concentration)\) were analyzed using stats4::lm() function in R, using calibration data provided Table 5.1 and Figure 3.1).

Analysis were performed using :

  • Linear function (Linear) - Linear trough 0 function (Linear 0)
    • Linear trough 0 and the highest concentration levels function (Linear 0 - Max)
    • Linear function with weights (1/Y and 1/X) (Linear weighed 1/X and Linear weighed 1/Y)

Figure 3.1: Calibration data

3.2 Response functions obtained

3.2.1 Regression analysis

The interactive table 3.1 shows the values obtained with regressions :

Table 3.1: Results of linear regressions performed
Methods Serie Intercept Slope AIC R²
Linear (LM) 1 -1372.3658 7566912 188.82827 0.9987755
Linear (LM) 2 -2972.4758 7166164 157.35429 0.9999733
Linear (LM) 3 -4450.9923 7101554 178.31360 0.9996262
Linear 0 (LM) 1 0.0000 7559371 186.84949 0.9991446
Linear 0 (LM) 2 0.0000 7149831 159.29751 0.9999694
Linear 0 (LM) 3 0.0000 7077097 177.10514 0.9997112
Linear 0-max (LM) 1 0.0000 7559800 52.44744 0.9991522
Linear 0-max (LM) 2 0.0000 7151585 43.89910 0.9999868
Linear 0-max (LM) 3 0.0000 7083672 48.34795 0.9998756
Linear weighed 1/X (LM) 1 -1495.9007 7569138 165.77039 0.9987966
Linear weighed 1/X (LM) 2 -2165.6005 7151626 156.85434 0.9995574
Linear weighed 1/X (LM) 3 -530.7419 7030919 166.86941 0.9984005
Linear weighed 1/Y (LM) 1 -1709.6429 7563307 165.04285 0.9986945
Linear weighed 1/Y (LM) 2 -2215.1444 7146698 159.77107 0.9991691
Linear weighed 1/Y (LM) 3 -575.2321 7018751 167.75477 0.9981079

3.2.2 Residues for each calibration curves at each levels

The residues obtained are shown in the interactive figure 3.2.

You can click or double-click on a legend to isolate or display a curve. You can zoom in on a part of the figure.

Figure 3.2: Relative bias calculated from regression

4 Validation

4.1 Using a linear calibration curve

4.1.1 Trueness and precision obtained

Trueness and precision are depicted on table 4.1 and 4.2

Table 4.1: Trueness and precision estimators and limits ( LIN )
Introduced concentrations NULL Mean calculated concentrations NULL Bias NULL Repeatability SD NULL Between series SD NULL Intermediate precision SD NULL Low limit of tolerance NULL High limit of tolerance NULL
0.0005 0.0007245 0.0002245 0.0001373 1.0e-07 0.0002658 0.0002380 0.0012109
0.0015 0.0016887 0.0001887 0.0002630 0.0e+00 0.0003047 0.0012306 0.0021468
0.0200 0.0195872 -0.0004128 0.0014254 6.0e-07 0.0016225 0.0171654 0.0220090
0.2000 0.1999305 -0.0000695 0.0114413 7.6e-06 0.0117701 0.1830348 0.2168261
Table 4.2: Trueness and precision estimators and limits ( LIN )
Introduced concentrations NULL Mean calculated concentrations NULL Bias (%) Recovery (%) CV repeatability (%) CV intermediate precision (%) Low limit of tolerance (%) High limit of tolerance (%) Results
0.0005 0.0007245 44.8950192 144.89502 27.459237 53.167504 -52.399408 142.189446 FAIL
0.0015 0.0016887 12.5798108 112.57981 17.534570 20.311394 -17.958278 43.117900 FAIL
0.0200 0.0195872 -2.0641339 97.93587 7.126778 8.112455 -14.173073 10.044805 FAIL
0.2000 0.1999305 -0.0347674 99.96523 5.720655 5.885032 -8.482581 8.413046 FAIL

4.1.2 Accuracy profile

Accuracy profile are shown Figure 4.1. The 𝛽-tolerance interval (blue lines) should be entirely within the acceptance limits (red dashed lines).

Figure 4.1: Accuracy profiles (red dashed line: acceptance limits, blue lines: 𝛽-tolerance intervals).

4.1.3 Linearity profile

Linearity profile are shown Figure 4.2. You should check that the black line (linear model) should be superimposed on the red dashed identity line.

Figure 4.2: Linearity profiles (red dashed line: identity line, black lines: linear regression lines, blue lines: 𝛽-tolerance intervals).

4.1.4 Linear regression

The p-value for the intercept is > 0.05 which is in favor of a non-significant value (PASS)

The p-value for the slope is < 0.05 which is in favor of a significant value (PASS)

Table 4.3: Results of the linear regression.
Estimate CI (lower) CI (upper) Std. Error t value Pr(>|t|)
(Intercept) 0.0000181 -0.0019935 0.0020297 0.0009994 0.0181474 0.986
x 0.9993616 0.9793459 1.0193772 0.0099437 100.5019077 <0.001 ***

4.1.5 Studentized Breusch-Pagan test for heteroskedasticity

The p-value < 0.05 which is in favor of heterosedasticity and standard deviations of a calculated concentrations as related to introduced concentrations, are non-constant (FAIL)

Result of the studentized Breusch-Pagan test
Test statistic df P value
5.044 1 0.02471 *

4.2 Using a linear forced trough 0 calibration curve

4.2.1 Trueness and precision obtained

Trueness and precision are depicted on table 4.1 and 4.2

Table 4.4: Trueness and precision estimators and limits ( LIN_0 )
Introduced concentrations NULL Mean calculated concentrations NULL Bias NULL Repeatability SD NULL Between series SD NULL Intermediate precision SD NULL Low limit of tolerance NULL High limit of tolerance NULL
0.0005 0.0003176 -0.0001824 0.0001376 0.0e+00 0.0001376 0.0001219 0.0005133
0.0015 0.0012839 -0.0002161 0.0002636 0.0e+00 0.0002636 0.0009091 0.0016587
0.0200 0.0192220 -0.0007780 0.0014294 8.0e-07 0.0016996 0.0166384 0.0218056
0.2000 0.1999679 -0.0000321 0.0114542 7.5e-06 0.0117785 0.1830626 0.2168732
Table 4.5: Trueness and precision estimators and limits ( LIN_0 )
Introduced concentrations NULL Mean calculated concentrations NULL Bias (%) Recovery (%) CV repeatability (%) CV intermediate precision (%) Low limit of tolerance (%) High limit of tolerance (%) Results
0.0005 0.0003176 -36.4882817 63.51172 27.525333 27.525333 -75.627610 2.651046 FAIL
0.0015 0.0012839 -14.4058966 85.59410 17.570894 17.570894 -39.390625 10.578832 FAIL
0.0200 0.0192220 -3.8899630 96.11004 7.146753 8.497999 -16.808075 9.028149 FAIL
0.2000 0.1999679 -0.0160589 99.98394 5.727123 5.889253 -8.468714 8.436596 FAIL

4.2.2 Accuracy profile

Accuracy profile are shown Figure 4.1. The 𝛽-tolerance interval (blue lines) should be entirely within the acceptance limits (red dashed lines).

Figure 4.3: Accuracy profiles (red dashed line: acceptance limits, blue lines: 𝛽-tolerance intervals).

4.2.3 Linearity profile

Linearity profile are shown Figure 4.2. You should check that the black line (linear model) should be superimposed on the red dashed identity line.

Figure 4.4: Linearity profiles (red dashed line: identity line, black lines: linear regression lines, blue lines: 𝛽-tolerance intervals).

4.2.4 Linear regression

The p-value for the intercept is > 0.05 which is in favor of a non-significant value (PASS)

The p-value for the slope is < 0.05 which is in favor of a significant value (PASS)

Table 4.6: Results of the linear regression.
Estimate CI (lower) CI (upper) Std. Error t value Pr(>|t|)
(Intercept) -0.0003905 -0.0024046 0.0016237 0.0010006 -0.3902286 0.698
x 1.0015911 0.9815504 1.0216318 0.0099561 100.6003838 <0.001 ***

4.2.5 Studentized Breusch-Pagan test for heteroskedasticity

The p-value < 0.05 which is in favor of heterosedasticity and standard deviations of a calculated concentrations as related to introduced concentrations, are non-constant (FAIL)

Result of the studentized Breusch-Pagan test
Test statistic df P value
5.05 1 0.02462 *

4.3 Using a linear calibration curve with the highest level only and forced trough 0

4.3.1 Trueness and precision obtained

Trueness and precision are depicted on table 4.1 and 4.2

Table 4.7: Trueness and precision estimators and limits ( LIN_0MAX )
Introduced concentrations NULL Mean calculated concentrations NULL Bias NULL Repeatability SD NULL Between series SD NULL Intermediate precision SD NULL Low limit of tolerance NULL High limit of tolerance NULL
0.0005 0.0003174 -0.0001826 0.0001376 0.0e+00 0.0001376 0.0001218 0.0005130
0.0015 0.0012834 -0.0002166 0.0002635 0.0e+00 0.0002635 0.0009088 0.0016580
0.0200 0.0192142 -0.0007858 0.0014286 9.0e-07 0.0017009 0.0166273 0.0218012
0.2000 0.1998863 -0.0001137 0.0114533 7.8e-06 0.0117893 0.1829597 0.2168130
Table 4.8: Trueness and precision estimators and limits ( LIN_0MAX )
Introduced concentrations NULL Mean calculated concentrations NULL Bias (%) Recovery (%) CV repeatability (%) CV intermediate precision (%) Low limit of tolerance (%) High limit of tolerance (%) Results
0.0005 0.0003174 -36.5154449 63.48456 27.511253 27.511253 -75.634752 2.603862 FAIL
0.0015 0.0012834 -14.4401297 85.55987 17.563399 17.563399 -39.414202 10.533942 FAIL
0.0200 0.0192142 -3.9288280 96.07117 7.142830 8.504340 -16.863670 9.006014 FAIL
0.2000 0.1998863 -0.0568324 99.94317 5.726666 5.894633 -8.520142 8.406478 FAIL

4.3.2 Accuracy profile

Accuracy profile are shown Figure 4.1. The 𝛽-tolerance interval (blue lines) should be entirely within the acceptance limits (red dashed lines).

Figure 4.5: Accuracy profiles (red dashed line: acceptance limits, blue lines: 𝛽-tolerance intervals).

4.3.3 Linearity profile

Linearity profile are shown Figure 4.2. You should check that the black line (linear model) should be superimposed on the red dashed identity line.

Figure 4.6: Linearity profiles (red dashed line: identity line, black lines: linear regression lines, blue lines: 𝛽-tolerance intervals).

4.3.4 Linear regression

The p-value for the intercept is > 0.05 which is in favor of a non-significant value (PASS)

The p-value for the slope is < 0.05 which is in favor of a significant value (PASS)

Table 4.9: Results of the linear regression.
Estimate CI (lower) CI (upper) Std. Error t value Pr(>|t|)
(Intercept) -0.0003903 -0.0024057 0.0016251 0.0010013 -0.3897947 0.698
x 1.0011826 0.9811289 1.0212362 0.0099626 100.4942902 <0.001 ***

4.3.5 Studentized Breusch-Pagan test for heteroskedasticity

The p-value < 0.05 which is in favor of heterosedasticity and standard deviations of a calculated concentrations as related to introduced concentrations, are non-constant (FAIL)

Result of the studentized Breusch-Pagan test
Test statistic df P value
5.025 1 0.02498 *

4.4 Using a weighed (1/Y) linear calibration curve

4.4.1 Trueness and precision obtained

Trueness and precision are depicted on table 4.1 and 4.2

Table 4.10: Trueness and precision estimators and limits ( LIN_1Y )
Introduced concentrations NULL Mean calculated concentrations NULL Bias NULL Repeatability SD NULL Between series SD NULL Intermediate precision SD NULL Low limit of tolerance NULL High limit of tolerance NULL
0.0005 0.0005245 0.0000245 0.0001382 0.0e+00 0.0001529 0.0002988 0.0007502
0.0015 0.0014933 -0.0000067 0.0002644 0.0e+00 0.0002845 0.0010780 0.0019085
0.0200 0.0194800 -0.0005200 0.0014350 7.0e-07 0.0016699 0.0169644 0.0219955
0.2000 0.2007199 0.0007199 0.0114500 6.2e-06 0.0117193 0.1839268 0.2175130
Table 4.11: Trueness and precision estimators and limits ( LIN_1Y )
Introduced concentrations NULL Mean calculated concentrations NULL Bias (%) Recovery (%) CV repeatability (%) CV intermediate precision (%) Low limit of tolerance (%) High limit of tolerance (%) Results
0.0005 0.0005245 4.8951205 104.89512 27.633709 30.588690 -40.246485 50.036726 FAIL
0.0015 0.0014933 -0.4488081 99.55119 17.627064 18.967374 -28.130321 27.232705 FAIL
0.0200 0.0194800 -2.6001706 97.39983 7.174992 8.349560 -15.177868 9.977527 FAIL
0.2000 0.2007199 0.3599375 100.35994 5.724989 5.859629 -8.036602 8.756477 FAIL

4.4.2 Accuracy profile

Accuracy profile are shown Figure 4.1. The 𝛽-tolerance interval (blue lines) should be entirely within the acceptance limits (red dashed lines).

Figure 4.7: Accuracy profiles (red dashed line: acceptance limits, blue lines: 𝛽-tolerance intervals).

4.4.3 Linearity profile

Linearity profile are shown Figure 4.2. You should check that the black line (linear model) should be superimposed on the red dashed identity line.

Figure 4.8: Linearity profiles (red dashed line: identity line, black lines: linear regression lines, blue lines: 𝛽-tolerance intervals).

4.4.4 Linear regression

The p-value for the intercept is > 0.05 which is in favor of a non-significant value (PASS)

The p-value for the slope is < 0.05 which is in favor of a significant value (PASS)

Table 4.12: Results of the linear regression.
Estimate CI (lower) CI (upper) Std. Error t value Pr(>|t|)
(Intercept) -0.0001857 -0.0021921 0.0018206 0.0009967 -0.1863452 0.853
x 1.0043267 0.9843637 1.0242897 0.0099176 101.2676087 <0.001 ***

4.4.5 Studentized Breusch-Pagan test for heteroskedasticity

The p-value < 0.05 which is in favor of heterosedasticity and standard deviations of a calculated concentrations as related to introduced concentrations, are non-constant (FAIL)

Result of the studentized Breusch-Pagan test
Test statistic df P value
5.346 1 0.02077 *

4.5 Using a weighed (1/X) linear calibration curve

4.5.1 Trueness and precision obtained

Trueness and precision are depicted on table 4.1 and 4.2

Table 4.13: Trueness and precision estimators and limits ( LIN_1X )
Introduced concentrations NULL Mean calculated concentrations NULL Bias NULL Repeatability SD NULL Between series SD NULL Intermediate precision SD NULL Low limit of tolerance NULL High limit of tolerance NULL
0.0005 0.0005101 0.0000101 0.0001380 0.0e+00 0.0001525 0.0002852 0.0007351
0.0015 0.0014779 -0.0000221 0.0002641 0.0e+00 0.0002842 0.0010631 0.0018927
0.0200 0.0194454 -0.0005546 0.0014333 7.0e-07 0.0016611 0.0169475 0.0219434
0.2000 0.2004922 0.0004922 0.0114412 5.9e-06 0.0116946 0.1837421 0.2172423
Table 4.14: Trueness and precision estimators and limits ( LIN_1X )
Introduced concentrations NULL Mean calculated concentrations NULL Bias (%) Recovery (%) CV repeatability (%) CV intermediate precision (%) Low limit of tolerance (%) High limit of tolerance (%) Results
0.0005 0.0005101 2.0241985 102.02420 27.599817 30.504080 -42.96475 47.013147 FAIL
0.0015 0.0014779 -1.4727244 98.52728 17.606452 18.948361 -29.12827 26.182819 FAIL
0.0200 0.0194454 -2.7727795 97.22722 7.166690 8.305595 -15.26270 9.717137 FAIL
0.2000 0.2004922 0.2460936 100.24609 5.720583 5.847318 -8.12897 8.621157 FAIL

4.5.2 Accuracy profile

Accuracy profile are shown Figure 4.1. The 𝛽-tolerance interval (blue lines) should be entirely within the acceptance limits (red dashed lines).

Figure 4.9: Accuracy profiles (red dashed line: acceptance limits, blue lines: 𝛽-tolerance intervals).

4.5.3 Linearity profile

Linearity profile are shown Figure 4.2. You should check that the black line (linear model) should be superimposed on the red dashed identity line.

Figure 4.10: Linearity profiles (red dashed line: identity line, black lines: linear regression lines, blue lines: 𝛽-tolerance intervals).

4.5.4 Linear regression

The p-value for the intercept is > 0.05 which is in favor of a non-significant value (PASS)

The p-value for the slope is < 0.05 which is in favor of a significant value (PASS)

Table 4.15: Results of the linear regression.
Estimate CI (lower) CI (upper) Std. Error t value Pr(>|t|)
(Intercept) -0.0001993 -0.0022020 0.0018034 0.0009949 -0.2003272 0.842
x 1.0032564 0.9833293 1.0231835 0.0098997 101.3418223 <0.001 ***

4.5.5 Studentized Breusch-Pagan test for heteroskedasticity

The p-value < 0.05 which is in favor of heterosedasticity and standard deviations of a calculated concentrations as related to introduced concentrations, are non-constant (FAIL)

Result of the studentized Breusch-Pagan test
Test statistic df P value
5.301 1 0.02131 *

5 Raw data

Table 5.1: Calibration standards raw data
ID TYPE SERIE SIGNAL CONC_LEVEL
CAL CAL 1 3057 0.0005
CAL CAL 1 11600 0.0015
CAL CAL 1 155620 0.0200
CAL CAL 1 1556003 0.2000
CAL CAL 1 1720 0.0005
CAL CAL 1 7430 0.0015
CAL CAL 1 145383 0.0200
CAL CAL 1 1467917 0.2000
CAL CAL 2 2144 0.0005
CAL CAL 2 5250 0.0015
CAL CAL 2 142242 0.0200
CAL CAL 2 1425120 0.2000
CAL CAL 2 1847 0.0005
CAL CAL 2 8487 0.0015
CAL CAL 2 137393 0.0200
CAL CAL 2 1435514 0.2000
CAL CAL 3 2744 0.0005
CAL CAL 3 10473 0.0015
CAL CAL 3 141074 0.0200
CAL CAL 3 1400930 0.2000
CAL CAL 3 3422 0.0005
CAL CAL 3 10554 0.0015
CAL CAL 3 115746 0.0200
CAL CAL 3 1432539 0.2000
Table 5.2: Validation standards raw data
ID TYPE SERIE SIGNAL CONC_LEVEL
VAL VAL 1 1544 0.0005
VAL VAL 1 9992 0.0015
VAL VAL 1 155683 0.0200
VAL VAL 1 1767753 0.2000
VAL VAL 1 1422 0.0005
VAL VAL 1 11212 0.0015
VAL VAL 1 154877 0.0200
VAL VAL 1 1540288 0.2000
VAL VAL 1 3661 0.0005
VAL VAL 1 11527 0.0015
VAL VAL 1 154062 0.0200
VAL VAL 1 1466979 0.2000
VAL VAL 1 2712 0.0005
VAL VAL 1 6007 0.0015
VAL VAL 1 153763 0.0200
VAL VAL 1 1466715 0.2000
VAL VAL 2 2593 0.0005
VAL VAL 2 10558 0.0015
VAL VAL 2 140350 0.0200
VAL VAL 2 1413294 0.2000
VAL VAL 2 2880 0.0005
VAL VAL 2 9174 0.0015
VAL VAL 2 110853 0.0200
VAL VAL 2 1425878 0.2000
VAL VAL 2 1316 0.0005
VAL VAL 2 9065 0.0015
VAL VAL 2 136750 0.0200
VAL VAL 2 1368431 0.2000
VAL VAL 2 1908 0.0005
VAL VAL 2 9657 0.0015
VAL VAL 2 130685 0.0200
VAL VAL 2 1332873 0.2000
VAL VAL 3 1498 0.0005
VAL VAL 3 7730 0.0015
VAL VAL 3 142770 0.0200
VAL VAL 3 1430404 0.2000
VAL VAL 3 3415 0.0005
VAL VAL 3 10179 0.0015
VAL VAL 3 141127 0.0200
VAL VAL 3 1409384 0.2000
VAL VAL 3 3423 0.0005
VAL VAL 3 10600 0.0015
VAL VAL 3 138551 0.0200
VAL VAL 3 1403500 0.2000
VAL VAL 3 1281 0.0005
VAL VAL 3 6198 0.0015
VAL VAL 3 117689 0.0200
VAL VAL 3 1411380 0.2000

6 Packages

Aphalo, Pedro J. 2022a. Ggpmisc: Miscellaneous Extensions to Ggplot2. https://CRAN.R-project.org/package=ggpmisc.

Aphalo, Pedro J. 2022b. Ggpp: Grammar Extensions to Ggplot2. https://CRAN.R-project.org/package=ggpp.

Dahl, David B., David Scott, Charles Roosen, Arni Magnusson, and Jonathan Swinton. 2019. Xtable: Export Tables to LaTeX or HTML. http://xtable.r-forge.r-project.org/.

DarĂłczi, Gergely, and Roman Tsegelskyi. 2022. Pander: An r Pandoc Writer. https://rapporter.github.io/pander/.

Dowle, Matt, and Arun Srinivasan. 2022. Data.table: Extension of ’Data.frame‘. https://CRAN.R-project.org/package=data.table.

Fox, John, and Sanford Weisberg. 2019. An R Companion to Applied Regression. Third. Thousand Oaks CA: Sage. https://socialsciences.mcmaster.ca/jfox/Books/Companion/.

Fox, John, Sanford Weisberg, and Brad Price. 2022a. Car: Companion to Applied Regression. https://CRAN.R-project.org/package=car.

Fox, John, Sanford Weisberg, and Brad Price. 2022b. carData: Companion to Applied Regression Data Sets. https://CRAN.R-project.org/package=carData.

Garnier, Simon. 2021. Viridis: Colorblind-Friendly Color Maps for r. https://CRAN.R-project.org/package=viridis.

Garnier, Simon.. 2022. viridisLite: Colorblind-Friendly Color Maps (Lite Version). https://CRAN.R-project.org/package=viridisLite.

Hofner, Benjamin. 2021. papeR: A Toolbox for Writing Pretty Papers and Reports. https://CRAN.R-project.org/package=papeR.

Hofner, Benjamin, and with contributions by many others. 2021. papeR: A Toolbox for Writing Pretty Papers and Reports. https://github.com/hofnerb/papeR.

Hothorn, Torsten, Achim Zeileis, Richard W. Farebrother, and Clint Cummins. 2022. Lmtest: Testing Linear Regression Models. https://CRAN.R-project.org/package=lmtest.

R Core Team. 2022. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

Sievert, Carson. 2020. Interactive Web-Based Data Visualization with r, Plotly, and Shiny. Chapman; Hall/CRC. https://plotly-r.com.

Sievert, Carson, Chris Parmer, Toby Hocking, Scott Chamberlain, Karthik Ram, Marianne Corvellec, and Pedro Despouy. 2022. Plotly: Create Interactive Web Graphics via Plotly.js. https://CRAN.R-project.org/package=plotly.

Wickham, Hadley. 2016. Ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York. https://ggplot2.tidyverse.org.

Wickham, Hadley, and Jennifer Bryan. 2022. Readxl: Read Excel Files. https://CRAN.R-project.org/package=readxl.

Wickham, Hadley, Winston Chang, Lionel Henry, Thomas Lin Pedersen, Kohske Takahashi, Claus Wilke, Kara Woo, Hiroaki Yutani, and Dewey Dunnington. 2022. Ggplot2: Create Elegant Data Visualisations Using the Grammar of Graphics. https://CRAN.R-project.org/package=ggplot2.

Wickham, Hadley, Romain François, Lionel Henry, and Kirill Mßller. 2022. Dplyr: A Grammar of Data Manipulation. https://CRAN.R-project.org/package=dplyr.

Zeileis, Achim, and Gabor Grothendieck. 2005. “Zoo: S3 Infrastructure for Regular and Irregular Time Series.” Journal of Statistical Software 14 (6): 1–27. https://doi.org/10.18637/jss.v014.i06.

Zeileis, Achim, Gabor Grothendieck, and Jeffrey A. Ryan. 2022. Zoo: S3 Infrastructure for Regular and Irregular Time Series (z’s Ordered Observations). https://zoo.R-Forge.R-project.org/.

Zeileis, Achim, and Torsten Hothorn. 2002. “Diagnostic Checking in Regression Relationships.” R News 2 (3): 7–10. https://CRAN.R-project.org/doc/Rnews/.

Zhu, Hao. 2021. kableExtra: Construct Complex Table with Kable and Pipe Syntax. https://CRAN.R-project.org/package=kableExtra.